Question: Simplify the following expression: $ p = \dfrac{1}{9} - \dfrac{8q}{-8q + 5} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-8q + 5}{-8q + 5}$ $ \dfrac{1}{9} \times \dfrac{-8q + 5}{-8q + 5} = \dfrac{-8q + 5}{-72q + 45} $ Multiply the second expression by $\dfrac{9}{9}$ $ \dfrac{8q}{-8q + 5} \times \dfrac{9}{9} = \dfrac{72q}{-72q + 45} $ Therefore $ p = \dfrac{-8q + 5}{-72q + 45} - \dfrac{72q}{-72q + 45} $ Now the expressions have the same denominator we can simply subtract the numerators: $p = \dfrac{-8q + 5 - 72q }{-72q + 45} $ Distribute the negative sign: $p = \dfrac{-8q + 5 - 72q}{-72q + 45}$ $p = \dfrac{-80q + 5}{-72q + 45}$ Simplify the expression by dividing the numerator and denominator by -1: $p = \dfrac{80q - 5}{72q - 45}$